If \([a] \in \mathrm {Cu}(A)\)  is compact, then zero is an isolated point in the spectrum of \(a\)  and the Proposition follows from Lemma REF .
If \(a\)  is not compact, then real rank zero implies the existence of a sequence \(q_1,q_2,\ldots \)  of projections in \(A \otimes \mathcal {K}\)  such that \(q_i \ne q_{i+1}\) , \([q_i] \ll [q_{i+1}]\) , and
\(\sup _i [q_i] = [a]\) 
Since \(A\)  is approximately divisible, the \(\mathrm {K}_0\) -group of \(A\)  is weakly unperforated. It follows that \([q_{i}]-[q_{i-1}] = [r_i]\)  for each \(i >1\)  and some projection \(0 \ne r_i \in A \otimes \mathcal {K}\) . Set \(r_1 = q_1\) . Apply Lemma REF  with \(q = r_1\)  and \(a\)  as in the present Proposition to find a projection \(p_1 \sim r_1\)  and positive element \(b_1\)  in \(\overline{aAa}\)  such that \(p_1 b_1 = 0\)  and
\(d_\tau (a) = d_\tau (p_1) + d_\tau (b_1), \ \forall \tau \in \mathrm {T}(A).\) 
Assume, inductively, that we have found in \(\overline{aAa}\)  mutually orthogonal projections \(p_i \sim r_i\) , \(1 \le i \le n\) , and a positive element \(b_n\)  orthogonal to each \(p_i\)  such that
\(d_\tau (a) = d_\tau \left( \oplus _{i=1}^n p_i \right) + d_\tau (b_n) = \sum _{i=1}^n d_\tau (p_i) + d_\tau (b_n).\) 
Applying Lemma REF  with \(a=b_n\)  and \(q=r_{n+1}\)  and labeling the resulting projection and positive element as \(p_{n+1}\)  and \(b_{n+1}\) , respectively, yields (REF ) above with \(n\)  replaced by \(n+1\) . Continuing inductively in this manner produces a sequence of mutually orthogonal projections \(p_1,p_2,\ldots \)  in \(\overline{aAa}\) .
Set
\(b = \bigoplus _{i=1}^\infty \frac{1}{2^i} p_i.\) 
Note that zero is an accumulation point of the spectrum of \(b\) , so that \(b\)  is not compact in \(\mathrm {Cu}(A)\) . Now
\(q_n \sim \oplus _{i=1}^n p_i \precsim b \precsim a\) 
by Lemma  so that
\(d_\tau (q_n) \le d_\tau (b) \le d_\tau (a), \ \forall \tau \in \mathrm {T}(A).\) 
Since \([a] = \sup _n[q_n]\)  in \(\mathrm {Cu}(A)\)  we then have
\(d_\tau (a) = \sup _n d_\tau (q_n) \le d_\tau (b) \le d_\tau (a), \ \forall \tau \in \mathrm {T}(A)\) 
and so
\(d_\tau (b) = d_\tau (a), \ \forall \tau \in \mathrm {T}(A).\) 
\(A\)  is approximately divisible, hence \(\mathcal {Z}\) -stable, and so has strict comparison of positive elements ([1]}, [2]}). Since neither \(a\)  nor \(b\)  is compact it follows that \([a]=[b]\)  in \(\mathrm {Cu}(A)\)  ([3]}).
Set
\(h(t) = (1-t)a +tb, \ t \in [0,1].\) 
Evidently, \([h(0)] = [h(1)] = [a]\) . Since \(h(t) \in \overline{aAa}\)  we have \([h(t)] \le [a]\) . On the other hand, for \(t \in (0,1)\) ,
\(a \sim (1-t)a \precsim (1-t)a + tb = h(t),\) 
whence \([h(t)] \ge [a]\)  also.
